When do skew-products exist?

Abstract

The classical skew-product decomposition of planar Brownian motionrepresents the process in polar coordinates as an autonomously Markovian part and an part that is an independent Brownian motion on the unit circle time-changed according to the part. Theorem 4 of L09 gives a broad generalization of this fact to a setting where there is a diffusion on a manifold $X$ with a distribution that is equivariant under the smooth action ofa Lie group $K$. Under appropriate conditions, there is a decomposition into an autonomously Markovian radial part that lives on the space of orbits of $K$ and an angular part that is an independent Brownian motion on the homogeneous space $K/M$, where $M$ is the isotropy subgroup of a point of $x$, that is time-changed with a time-change that is adapted to the filtration of the part. We present two apparent counterexamples to Theorem 4 of L09. In the first counterexample the part is not a time-change of any Brownian motionon $K/M$, whereas in the second counterexample the part is the time-change of a Brownian motion on $K/M$ but this Brownian motion is not independent of the part. In both of these examples $K/M$ has dimension $1$. The statement and proof of Theorem 4 in L09 remain valid when $K/M$ has dimension greater than $1$. Our examples raise the question of what conditions lead to the usual sort of skew-product decomposition when $K/M$ has dimension $1$ and what conditions lead to there being no decomposition at all or one in which the part is a time-changed Brownian motion but this Brownian motion is not independent of the part.